📊 Inferential Statistics & Normal Distribution
Z-Score to P-Value Probability Calculator
Convert raw values to Standard Normal Z-scores ($Z = \frac{x - \mu}{\sigma}$), calculate left-tail, right-tail, and two-tailed P-values, and determine statistical significance ($\alpha = 0.05, 0.01$).
90% Confidence ($Z = 1.645$)
95% Confidence ($Z = 1.960, p = 0.05$)
99% Confidence ($Z = 2.576, p = 0.01$)
3-Sigma Outlier ($Z = 3.0$)
Population & Sample Inputs
Standardized Z-Score
Z = +1.000
Left Tail: $P(Z < z)$
0.8413
Right Tail: $P(Z > z)$
0.1587
Two-Tailed P-Value
p = 0.3173
💡 Significance Verdict:
At $Z = +1.00$, the two-tailed p-value is 0.3173 ($p > 0.05$), failing to reject the null hypothesis at standard 95% confidence level.
The Standard Normal Distribution
The Gaussian bell curve transforms raw data into standard deviation units:
- Z-Score: $$Z = \frac{x - \mu}{\sigma}$$
- Cumulative Distribution Function ($\Phi(Z)$): $$\Phi(Z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^Z e^{-\frac{t^2}{2}} \, dt$$
- Two-Tailed P-Value: $$p = 2 \times (1 - \Phi(|Z|))$$